Pu, Hung-Yi
Institute of Astronomy, National Tsing-Hua University
The Effects of Photon Path Bending on the Observed
Pulse Profile and Spectra of Surface Thermal Emission
from Neutron Stars
Spectra calculation:
∫I’ν(t) cosθ’ dΩ’
( Include photon path bending )
Motivation:
Application: Determining the inclination and viewing angles for Radio-quiet neutron stars from their X-ray thermal emission
Neutron stars with X-ray thermal emission Iν= Planck function
Consider Limb-darkening Iν=
(require )
The dependence of model spectra and pulse profiles on the
input parameters
ζ: the viewing angle α: the inclination angle
Light curveswith different temperature distribution (I)
Light curveswith different temperature distribution (II)
Light curveswith different temperature distribution (III)
Light curveswith and without limb-darkening
Dash: without limb-darkening
Dot: with limb-darkening
Spectrawith different magnetic field from
, Tp=2 x106K
Solid curve: Planck function (with suitable normalization) of temperature 2 x106 K
Determining the
inclination angle and
viewing angle of neutron stars from their X-ray thermal emissions
Observation:
Blackbody Best Fit Temperature(AσT4=4πd2F)
1)Observed Flux2)Pulse Fraction
Input parameters:1) Hot Spot Size2) M / R3) α and ζ
Inferred possible range of α and ζ
Consistent?
Model fluxModel Pulsed fraction
Yeah
No
Observational properties of RX J0806.4-4123
Apply T hot spot = 95.6 eVM/R=0.2
Different hot spot size:12,13,15,16 from top to down(in unit of canonical polar cap size,~0.25 degree)
Left: computed flux{Flux/28.8x10-13}={0.9,1.0,1.1,1.2}Right: computed pulsed fraction{0.05,0.06,0.07,0.1,0.2,0.3,0.4,0.5}
The shapes of computed pulsed fraction contours are not change too much for fixed mass-to-radius ratio
Computed pulsed fraction contoursfor different mass-to-radius ratio
M / R=0.01
M / R=0.1
M / R=0.2
M / R=0.3
Sum of computed RX J0806.4-4123 pulsed fraction contours
with values of 6%
M / R=0.01
M / R=0.1
M / R=0.2
M / R=0.3
Pulse profilese.x. M/R= 0.1, hot spot size equal to 22 times of the
canonical polar cap sizes
117 Radio pulsar geometries reported by Rankin ( 1993 )
Distribution of Radio pulsars
in α-ζ plane
M / R=0.01
M / R=0.1
M / R=0.2
M / R=0.3
RX J0420.0-5022M/R=0.01 0.1
0.2 0.3
{0.10, 0.12, 0.15, 0.2, 0.3}
RX J0720.4-3125
M/R=0.01 0.1
0.2 0.3
{0.06, 0.11, 0.15, 0.2, 0.3}
Summary
• Pulse fraction in general decreases as the gravity increases; Beaming effect in general increases the pulse fraction.
• The inferred possible geometry range is broad in the α-ζ plane and we cannot tell if the distribution for radio-quiet neutron stars and radio pulsars are different.
• Our computed result depends on the beaming pattern we use.
Thank you!
ζ:
the viewing angle
α: the inclination angle
Lyne 1998
Rotation Axis
Magnetic Axis
Hot Spot
∫I’ν(t) cosθ’ dΩ’Calculate (specific)Observed Flux:
dΩ’
Distant Observer
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